Question

A computer programmer makes $60,000 in her first year of working at a company. She gets a 10 percent pay raise every year. Create a geometric series model for how much she makes in his first 5 years of working.(1 point)
Responses

∑5n=360,000(1.1)n−1
sigma-summation underscript n equals 3 overscript 5 End Scripts 60,000 left parenthesis 1.1 right parenthesis superscript n minus 1 baseline

∑5n=560,000(1.1)n
sigma-summation underscript n equals 5 overscript 5 End Scripts 60,000 left parenthesis 1.1 right parenthesis superscript n baseline

∑5n=160,000(1.1)n−1
sigma-summation underscript n equals 1 overscript 5 End Scripts 60,000 left parenthesis 1.1 right parenthesis superscript n minus 1 baseline

∑5n=160,000(0.1)n−1

Answer 1

To model the salary of the programmer over the first 5 years, we can set up a geometric series where the first term \( a = 60,000 \) and the common ratio \( r = 1.1 \) (due to the 10% pay raise each year).

The salary for each year can be expressed as follows:

- Year 1: \( 60,000 \)
- Year 2: \( 60,000 \times 1.1 \)
- Year 3: \( 60,000 \times (1.1)^2 \)
- Year 4: \( 60,000 \times (1.1)^3 \)
- Year 5: \( 60,000 \times (1.1)^4 \)

The total amount she makes in the first 5 years is given by the sum of these terms, which can be expressed as:

\[
S = 60,000 + 60,000 \times 1.1 + 60,000 \times (1.1)^2 + 60,000 \times (1.1)^3 + 60,000 \times (1.1)^4
\]

This can be rewritten in summation notation as:

\[
S = \sum_{n=1}^{5} 60,000 \times (1.1)^{n-1}
\]

Looking at your provided options, the correct representation of the series is:

\[
\sum_{n=1}^{5} 60,000 \left(1.1\right)^{n-1}
\]

So, the correct answer is:

**sigma-summation underscript n equals 1 overscript 5 End Scripts 60,000 left parenthesis 1.1 right parenthesis superscript n minus 1 baseline**